I am reading Li's Geometric Analysis on upper and lower bounds for the heat kernel in the presence of a lower Ricci curvature bound, and I am hoping to gain some intuition for why they hold and why Ricci control is necessary.
I can gain some intuition from thinking about the space forms--heat dissipates faster on hyperbolic space and slower on the sphere, presumably because in the first case there is "more volume to flow out to" and in the second case there is less. I guess a narrow neck would also impede heat flow and this would correspond to very negative Ricci curvature.
Is this all that's going on? Are there other examples the would be useful to think about?